Lesson 7.1 Angles of Polygons
Essential Question: How can I find the sum of the measures of the interior angles of a polygon?
Polygon A plane figure made of three or more segments (sides). Each side intersects exactly two other sides at their endpoints. Polygons are named by vertices in consecutive order, going CW or CCW.
Diagonals in a Polygon A segment that joins two non-consecutive vertices. The diagonals from one vertex divide a polygon into triangles.
Interior Angle Sum of a Triangle m 1 = m 4 4 3 5 m 2 = m 5 180 1 2 m 4 m 1 + m 5 m 2 + m 3 = 180 1 180
Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180
Interiore Angle Sum in a Quadrilateral 1 4 2 3 6 2 s 360 m 1 + m 2 + m 3 = 180 5 m 4 + m 5 + m 6 = 180 m 1 + m 4 + m 2 + m 5 + m 3 + m 6 = 360
Polygon Interor Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360
Interior Angle Sum in a Pentagon From this vertex, how many diagonals are there? 2
Angle Sum in a Pentagon How many triangles are there? 3 And what is the sum of the angles of each triangle? 180 180 180 180
Angle Sum in a Pentagon 3 s 540 180 So what is the sum of the interior angles of a pentagon? 180 180 3 180 = 540
Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540
Interior Angle Sum in a Hexagon How many diagonals from this vertex? 3 How many triangles are formed? 4 The sum of the angles is? 4 180 = 720 4 s 720
Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540 Hexagon 6 4 720
Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540 Hexagon 6 4 720 Octagon 8 6 1080
What s the pattern? A polygon with n sides can be divided into how many triangles? n 2 The sum of the angles then is? 180(n 2)
Polygon Interior Angle Sums Polygon Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540 Hexagon 6 4 720 Octagon 8 6 1080 n-gon n n 2 180(n 2)
Theorem 7.1 Polygon Interior Angles Theorem The sum of the interior angles of a convex n- gon is: 180 ( n 2) memorize this!
Example 1: Find the sum of the interior angles of a polygon with 14 sides. 180(14 2) = 180(12) = 2160
Example 2: The sum of the interior angles of a polygon is 2700. How many sides does the polygon have? 17 180(n 2) = 2700 180n 360 = 2700 180n = 3060 n = 17
Example 3: The sum of the interior angles of a polygon is 1620. How many sides does the polygon have? 11 180(n 2) = 1620 180n 360 = 1620 180n = 1980 n = 11
Example 4: The sum of the interior angles of a polygon is 1380. How many sides does the polygon have? This is not a polygon. 180(n 2) = 1380 180n 360 = 1380 180n = 1740 n = 9.66666 Why must this be a whole number?
7.1 Corollary to the Polygon Interior Angles Theorem The sum of the interior angles of a quadrilateral is 360. 1 4 2 3 1 + 2 + 3 + 4 = 360
Example 5: Solve for x. x 55 x x + x + 55 + 55 = 360 2x + 110 = 360 2x = 250 x = 125
Your Turn Find the value of x in the diagram. x + 108 + 121 + 59 = 360 x + 288 = 360 x = 72
Regular Polygons
Regular Polygon All sides congruent All angles congruent The Sum of the interior angles is 180 (n 2) Since the angles are congruent, the measure of each interior angle in a regular polygon is E I = 180 (n 2) n
Example 6: Find the measure of each angle of a regular pentagon. 108 180(5 2) 180(3) 5 5 108 108 108 540 108 5 108 108
Example 7: Each angle of a regular polygon measures 160. How many sides does the polygon have? 180( n 2) 160 n 180n 360 160n 20n 360 n 18
Example 8: A home plate for a baseball field is shown. a. Is the polygon regular? Explain your reasoning. The polygon is not equilateral or equiangular. So, the polygon is not regular.
Example 8: b. Find the measures of C and E. 180 (n 2) = 180 (5 2) = 540 x + x + 90 + 90 + 90 = 540 2x + 270 = 540 2x = 270 x = 135 Therefore, C= 135 and E= 135
The Sum of the Exterior Angles This always means using one exterior angle at each vertex. But not this one. This angle or this angle.
The Sum of the Exterior Angles Extend only ONE side at each vertex. Exterior Angle Exterior Angle Exterior Angle
Theorem 7.2 Polygon Exterior Angles Theorem The sum of the exterior angles of any polygon, one angle at each vertex, is 360. S E = 360 m 1 + m 2 + + m n = 360
Example 9: Find the value of x in the diagram. x + 2x + 89 + 67 = 360 3x + 156 = 360 3x = 204 x = 68
Corollary The measure of an exterior angle of a regular polygon with n sides is E = 360 E n
Example 10: Find the measure of an exterior angle of a regular 40-gon. Solution: 360/40 = 9
Example 15: The trampoline shown is shaped like a regular dodecagon. a. Find the measure of each interior angle. 180 ( n 2) n 180(12 2) = 12 = 180(10) 12 = 1800 = 150 12
Example 15: The trampoline shown is shaped like a regular dodecagon. b. Find the measure of each exterior angle. 360 12 = 30
Summary The sum of the interior angles of an n-gon is 180(n 2). The sum of the exterior angles of any polygon is 360. The measure of an interior angle of a regular polygon is 180(n 2) n The measure of an exterior angle of a regular polygon is 360 n..